Triplet Spin Delocalization and Temperature Dependence for Adiabatic and Non-Adiabatic Z–E Isomerization Pathways in Azoarenes

Abstract

Photoswitching molecules like the azoarenes have myriad potential applications, ranging from energy storage to targeted drug delivery. Upon irradiation, azoarenes convert from an E-form to a Z-form. The Z-form can thermally revert to the E-form through a classical-adiabatic or a triplet-assisted rotational mechanism. We show that strategic placements of heteroatoms and substituents can modulate spin delocalization in the triplet state, thereby tuning the S₀–T₁–S₀ crossing points. Increased spin delocalization lowers the crossing point and shortens the thermal half-life of the Z-form. Computed spin densities at the diazo N atoms, along with computed S₀–T₁–S₀ crossing points, activation entropies, and rate constants for 15 azoarenes are compared with available experimental data. There is evidence that some of the studied Z–E isomerizations proceed simultaneously via a classical-adiabatic and a triplet-assisted mechanism. The ratio of the two reaction pathways is temperature-dependent. Triplet spin density at the diazo N atoms is recognized as a useful indicator for predicting the thermal half-lives of these species.

Graphical Abstract

INTRODUCTION

Azoarenes are photoswitching molecules that can convert between two conformers: a low energy E-form, which converts to a metastable Z-form by irradiation, and a Z-form that thermally relaxes back to the E-form (Figure 1a).^1-4 The thermal half-life of the Z-form is a key parameter for photoswitching applications and can vary widely—from seconds to days or even years—depending on the molecular structure. Short half-lives (ranging from seconds to minutes) align with the time scale of biological events,^5,6 while longer half-lives are advantageous for energy storage.^7,8 Many azoarene derivatives have been prepared^9-18 and some structure–reactivity relationships have been proposed to modulate the thermal half-lives of the Z-form. For example, C–H···π^9,10 and lone pair···π¹¹ interactions have been suggested to stabilize Z-form structures that adopt T-shaped geometries at the rotational transition state, thereby extending the thermal half-lives. Adrion and Lopez examined thermal relaxation through an inversion pathway and found that cross-conjugated azofurans and azothiophenes exhibit longer thermal half-lives than their noncross-conjugated counterparts.^15,18 Similarly, bridged azoarenes show a range of thermal half-lives depending on the bridge length and the resulting macrocycle strain.^19-21 Despite these variations, most studies assume that Z–E isomerization proceeds through a single-state mechanism—either by rotation around the N═N bond, or inplane inversion around a N center, or a combination of both.

Figure 1.

a) Photochemical E–Z and thermal Z–E isomerization of azobenzene. b) Schematic energy diagram showing Z–E isomerization via a triplet-assisted rotational pathway. Minimum energy crossing points (MECP), M₁ and M₂, mark the S₀–T₁–S₀ crossings.

The thermal Z–E isomerization of azobenzene proceeds through a multistate nonadiabatic S₀–T₁–S₀ mechanism—a pathway that has been largely overlooked for the past two decades. In a 2004 computational study, Cembran et al.^22,23 demonstrated that the most favorable thermal isomerization route involves rotation around the N═N bond and a crossing with the T₁ potential energy surface.^22,23 CASPT2 calculations revealed that, at the geometries of the rotational transition state, the T₁ state lies below the S₀ state^22,23 (as illustrated in Figure 1b). The rotational energy profile of the S₀ state intersects the T₁ surface at two minimum energy crossing points (MECP): M₁ is the S₀ to T₁ crossing, and M₂ is the T₁ to S₀ crossing.

In 2019, Magnussen, Tuczek, Herges and co-workers reported the first experimental proof supporting a multistate thermal relaxation for azobenzene.²⁴ They observed tremendous Z–E isomerization rate accelerations (up to 1300-fold) for a series of functionalized azobenzene on bulk gold surfaces compared to rates measured in solution. The authors attributed this acceleration to enhanced quantum mechanical transition probabilities associated with a S₀–T₁–S₀ crossing and noted that “the fundamental change in the cis-to-trans reaction observed [ …] can be correlated with the degree of electronic coupling to the [gold] surface.” The Z–E isomerization rates were found to correlate with the spin-density distribution in the triplet state of the functionalized azobenzenes. These findings suggest that two factors govern the rate of thermal Z–E isomerization: the transmission coefficient (i.e., the probability of a S₀–T₁–S₀ crossing), which is temperature independent, and the energy level of the triplet surface and extent of triplet state population, which is temperature dependent.

A number of azobenzenes, including the parent system, exhibit strongly negative activation entropies (ΔS^‡) that are not consistent with a classical-adiabatic mechanism. Adiabatic inversion as well as rotation at the azo group on the singlet ground-state hypersurface, in contrast, are predicted to have rather small positive values for ΔS^‡. Later, Reimann et al. solved this so-called “entropy puzzle” of azobenzene, demonstrating that calculated and experimental Eyring parameters for the thermal relaxation of azobenzene match only when a nonadiabatic mechanism is considered.^25,26 Expanding on this finding, Axelrod et al. considered a nonadiabatic rotational mechanism and applied machine-learning methods to predict the thermal half-lives of 19,000 azoarenes.²⁷

Spin-forbidden reactions, such as the triplet-assisted rotational mechanism shown in Figure 1b, are governed by the nonadiabatic transition state theory (NA-TST). In NA-TST, the thermal reaction barrier is defined by the MECPs between potential energy surfaces of different spin multiplicities (Figure 1b). Surface crossing becomes feasible when the spin–orbit coupling (SOC) is sufficiently large typically in the range of 15–30 cm⁻¹ or higher, compared to ~1 cm⁻¹ for most organic compounds.^28-30 Singer et al. emphasized that the magnitude of SOC plays a critical role in determining the likelihood of a multistate rotational pathway. Obviously, not all Z–E isomerizations of azoarenes proceed exclusively via a nonadiabatic S₀–T₁–S₀ reaction pathway.³¹ In phenylazo-1,3,5-trimethylpyrazole, for example, the adiabatic inversion and nonadiabatic S₀–T₁–S₀ mechanism proceed simultaneously. Without accounting for the inversion mechanism and including both mechanisms, the half-lives are not accurately predicted.³¹

Can the thermal half-life of the Z-form be modulated by tuning the T₁ state potential energy surface? And if so, can the spin density at the diazo linker serve as an effective predictive indicator for the thermal half-lives? Recognizing that the nonadiabatic S₀–T₁–S₀ mechanism is an important pathway for the Z–E isomerization of azoarenes, we hypothesized that strategically introducing heteroatoms and substituents into the arene fragments could enhance or reduce spin delocalization. This, in turn, would stabilize or destabilize the T₁ energy surface, leading to faster or slower Z–E isomerization rates.

RESULTS AND DISCUSSION

We investigated the Z–E isomerization mechanisms of 15 azoarenes (Scheme 1). Activation barriers for the adiabatic inversion, adiabatic rotation, and the S₀–T₁–S₀ crossing points were computed at CASPT2/ANO-RCC-VTZP//(U)PBE0-D3(BJ)/def2-TZVPP. These data were then used to calculate rate constants (k) and activation entropies (ΔS^‡) for each mechanism, and the results were compared to available experimental data. While some compounds were found to undergo an exclusive nonadiabatic Z–E isomerization, others exhibited competing pathways. We further examined the temperature dependence of Z–E isomerization pathways and discuss its implications for the competition between adiabatic and nonadiabatic mechanisms. Finally, we identify spin density at the linker nitrogen atoms as a predictive indicator for estimating trends in the Z–E isomerization half-lives of azoarenes. These findings were applied to rationalize previously reported experimental trends. For example, He et al.³² observed that the azobispyrazole isomers, 3–5, exhibit markedly different half-lives ranging from 0.3 days (3), 72 days (4), to 681 days (5) depending on how the pyrazole rings are connected to the N═N linker. Similarly, Heindl and Wenger showed that the thermal half-life of phenylazothiophene 6 (t_1/2 ~ 7.1 h) can be tuned by substitution, as illustrated by the measured half-lives of 7p (t_1/2 ~ 0.5 h) vs 7m (t_1/2 ~ 7 h) and 8p (t_1/2 ~ 6.4 h) vs 8m (t_1/2 ~ 17.7 h).¹²

Scheme 1.

Azoarene Compounds Studied

Two key factors must be considered when evaluating the nonadiabatic rotation mechanism: 1) The transmission coefficient (i.e., the probability of a S₀–T₁–S₀ crossing) must be sufficiently large for this mechanism to occur. This factor is almost temperature-independent and can be estimated by computing the spin–orbit coupling matrix elements (SOCME) at the M₁ and M₂ crossing points. For all studied azoarenes, SOCME values at the M₁ and M₂ crossing points³¹ range from 6.9 to 22.8 cm⁻¹, which is significantly higher than the cf. 1 cm⁻¹ typical for organic compounds and can influence the reaction rate (see Figure S11). 2) Thermal population of the crossing trajectories at M₁ and M₂ increases with temperature. The energy levels of the M₁ and M₂ crossing points are influenced by triplet-state stabilization: lower energy M₁ and M₂ crossings correspond to faster Z–E isomerization rates, while higher energy M₁ and M₂ crossings lead to slower isomerization rates. M₁ and M₂ energies were located via minimum energy crossing point optimizations at (U)PBE0-D3(BJ)/def2-TZVPP, followed by single-point energy calculations at the CASPT2/ANO-RCC-VTZP level. Details for active space selection are provided in the SI.

Computed activation barriers for the adiabatic inversion (ΔE^‡_inv), adiabatic rotation (ΔE^‡_rot), and the S₀–T₁–S₀ crossing points (M₁ and M₂) were obtained at the CASPT2/ANO-RCC-VTZP level and summarized in Table 1. Across all compounds studied, the M₁ and M₂ energies are consistently lower than the activation barriers for both adiabatic mechanisms. Although the inversion pathway was previously considered competitive,^33-37 the computed activation barriers for adiabatic rotation are lower. Rate constants for the nonadiabatic (k_NA), adiabatic rotation (k_rot), and adiabatic inversion (k_inv) mechanisms were estimated using Eyring’s transition state theory and compared to available experimental rate constants (k_exp) (see details in the SI).^12,25,32 The k_NA values were calculated using the Eyring equation with an added transmission coefficient to account for spin–orbit coupling. As shown in Table 1, experimental k_exp constants match more closely with k_NA and k_rot, while k_inv significantly underestimate the observed rates by several orders of magnitude.

Table 1.

Computed M₁ and M₂ Values As Well As Activation Barriers for the Adiabatic Rotation (ΔE^‡_rot) and Adiabatic Inversion (ΔE^‡_inv) Pathways (relative to the Z form) at CASPT2/ANO-RCC-VTZP//(U)PBE0-D3(BJ)/def2-TZVPP (in kcal/mol)^a

	M1	M2	ΔE‡ rot	ΔE‡ inv	ΔS‡ NA	ΔS‡rot	ΔS‡inve	ΔS‡exp	kNA	krot	kinv e	kexp
Azb	22.0(25.3)b	21.0(22.9)b	28.0(30.4)b	30.6(31.1)b	−33.8	0.0	0.3	−38.0f	1.1 × 10−5	1.9 × 10−8	2.3 × 10−10	1.9 × 10−6i
1N	19.7	17.8	22.0	30.6	−45.1	0.0	0.3	−	1.2 × 10−4	4.6 × 10−4	2.3 × 10−10	−
2N	23.5	24.4	30.4	36.8	−38.8	−0.1	0.2	−	1.0 × 10−7	3.2 × 10−10	6.6 × 10−15	−
1O	16.5	15.5	18.7	26.6	−49.6	−0.2	−0.2	−	1.8 × 10−2	1.2 × 10−1	2.0 × 10−7	−
2O	24.1	23.3	29.2	36.5	−39.1	−0.1	0.1	−	1.7 × 10−7	2.5 × 10−9	1.1 × 10−14	−
1S	20.2	18.3	21.9	29.8	−	0.0	0.1	−	−	5.5 × 10−4	8.9 × 10−10	−
2S	24.8	23.4	30.8	35.2	−44.6	0.2	0.0	−	6.6 × 10−8	1.6 × 10−10	9.8 × 10−14	5.7 × 10−7j
3	21.9	20.6	26.5	31.9	−41.3	0.0	0.0	−43.9g	4.7 × 10−6	2.3 × 10−7	2.6 × 10−11	2.7 × 10−5k
4	25.4	23.0	31.7	33.5	−32.8	0.0	0.0	−55.4g	3.2 × 10−8	3.6 × 10−11	1.7 × 10−12	1.1 × 10−7k
5	26.1	24.2	31.9	36.8	−35.1	0.0	0.2	−41.2g	7.9 × 10−9	2.6 × 10−11	6.6 × 10−15	1.2 × 10−8k
6	22.2	21.2	26.9	32.9c	−41.7	−0.2	0.2	−20.9h	3.2 × 10−6	1.2 × 10−7	4.7 × 10−12	2.7 × 10−5l
7p	22.1	20.1	25.2	32.3c	−39.7	−0.6	−0.2	−12.5h	3.9 × 10−6	2.1 × 10−6	1.3 × 10−11	3.9 × 10−4l
7m	23.3	21.6	26.7	33.0c	−43.2	0.2	−0.1	−12.5h	3.2 × 10−7	1.7 × 10−7	4.0 × 10−12	2.8 × 10−5l
8p	23.7	20.3	−	30.6d	−37.5	−	0.2	−12.5h	3.0 × 10−7	−	2.3 × 10−10	3.0 × 10−5l
8m	24.5	22.1	26.7	33.1c	−42.9	0.2	0.1	−33.4h	3.9 × 10−8	1.6 × 10−7	3.4 × 10−12	1.1 × 10−5l

^aComputed activation entropies for triplet-assisted (ΔS^‡_NA), rotation (ΔS^‡_rot), and inversion (ΔS^‡_inv) pathways are compared to available experimental values (ΔS^‡_exp) (in J/molK). Estimated rate constants at 25 °C for triplet-assisted (k_NA), rotation (k_rot), and inversion (k_inv) pathways are compared to available experimental data (k_exp).

^bTheoretical best estimate energies as reported in ref 25. Inversion barrier around the C—N=N angle connected to the

^cthiophene or

^dbenzene ring.

^eSince inversion for 6 to 8 can occur at the benzyl ring or thiophene ring, the computed activation entropies are based on the fastest rate constants (k_inv).

^fData reported in ref 25.

^gDerived from data reported in ref 32.

^hDerived from data reported in ref 12.

ⁱRate constant measured in n-heptane (ref 39).

^jRate constant measured at 22 °C in DMSO (ref 38).

^kRate constant measured at 25 °C in DMSO (ref 32).

^lRate constant measured at 20 °C in acetonitrile (ref 12).

Computed activation entropies for the triplet-assisted rotation (ΔS^‡_NA), adiabatic rotation (ΔS^‡_rot), and adiabatic inversion (ΔS^‡_inv) pathways were compared to available experimental values (ΔS^‡_exp) (see Table 1 and Table S4).^12,25,32 As expected for a nonadiabatic process, all ΔS^‡_NA values are significantly negative (ranging from −32.8 to −49.6 J/mol·K). In contrast, the computed adiabatic values are near zero: ΔS^‡_rot (−0.6 to +0.2 J/mol·K) and ΔS^‡_inv (−0.2 to +0.3 J/mol·K). Comparisons with experimentally derived activation entropies suggest that several compounds, including Azb (ΔS^‡exp = −38.0 J/mol·K),²⁵ 3 (−43.9 J/mol K),³² 4 (−55.4 J/mol·K),³² and 5 (−41.2 J/mol·K),³² likely undergo Z–E isomerization via nonadiabatic rotation. In contrast, the smaller ΔS^‡_exp values for 6 (−20.9 J/mol·K),¹² 7p (−12.5 J/mol·K),¹² 7m (−12.5 J/mol·K),¹² 8p (−12.5 J/mol·K),¹² and 8m (−33.4 J/mol·K)¹² suggest a mixture of adiabatic and nonadiabatic pathways.

We next explore how heteroatoms can be strategically placed within the azoarene skeleton to modulate the T₁ state energies. Resonance structures of the triplet states for two azobispyrrole isomers, 1N and 2N, shown in Figure 2a, are instructive. In 1N, the triplet spin delocalizes to the C3 and C5 positions of the pyrrole rings. But in 2N, triplet spin delocalizes only to the C2 position. This difference is supported by natural bond orbital (NBO) calculations, which show higher spin densities at the carbon positions marked with a “dot” in Figure 2. Notably, the diazo N atoms of 1N exhibit lower spin density (0.59) than those of 2N (0.72), indicating greater spin delocalization in 1N and a more stabilized T₁ state potential energy surface. Indeed, relaxed rotational potential energy plots at the CASPT2/ANO-RCC-VTZP//(U)PBE0-D3(BJ)/def2-TZVPP level for 1N (orange line) and 2N (purple line) show a lower T₁ state potential energy surface and lower MECPs for 1N (Figure 2b). Solid lines represent the S₀ state, and dotted lines represent the T₁ state. Since 1N exhibits a more stabilized triplet state, the minimum energy crossing points also are lower: M₁ = 19.7 kcal/mol and M₂ = 17.8 kcal/mol, compared to M₁ = 23.5 kcal/mol and M₂ = 24.4 kcal/mol for 2N.

Figure 2.

a) Representative resonance structures and computed spin densities for the triplet state minima (top value) and singlet state rotational transition state structures (bottom value, in parentheses) of 1N, 2N, and 6 (only values larger than 0.2 are shown. b) Relaxed rotational energy profiles for 1N and 2N at CASPT2/ANO-RCC-VTZP//(U)PBE0-D3(BJ)/def2-TZVPP. Solid (and dotted) lines indicate the S₀ (and T₁) energy profiles.

Differences in spin delocalization help explain why cross-conjugated azoarenes (e.g., 2N) generally exhibit longer thermal half-lives than their non-cross-conjugated counterparts (e.g., 1N).^15,18 Non-cross-conjugated structures have more delocalized triplet spin, which lowers the T₁ energy surface and the S₀–T₁–S₀ crossing points, resulting in shorter thermal half-lives. In contrast, cross-conjugated structures exhibit less spin delocalization, leading to a higher T₁ state surface and higher energy crossing points, thus longer thermal half-lives. This trend is consistent across analogous azobisfuran (1O vs 2O) and azobisthiophene (1S vs 2S) isomers, as supported by the computed spin densities and M₁ and M₂ values (Figure S10).

Importantly, both the singlet and triplet state energy surfaces are influenced by diradical delocalization. Specifically, the energy of the singlet state profile (close to the transition state region) is largely affected by the extent of diradical delocalization; more delocalization in 1N (i.e., lower rotational barrier, solid orange line) and less delocalization in 2N (i.e., higher rotational barrier, solid purple line) (Figure 2b). Computed spin densities for the diradicaloid transition state structures of 1N and 2N (Figure 2a, values in parentheses) show the same trends as that computed for the triplet minima, indicating a more delocalized 1N. Thus, both the singlet and triplet energy profiles for 1N are lower in energy, while those for 2N are higher in energy.

We note important qualitative differences when comparing the singlet and triplet profiles of 1N vs 2N. As shown in Figure 2b, 2N likely undergoes an exclusive nonadiabatic rotational mechanism. Note the much lower triplet energy compared to the singlet energy at the rotational transition state and triplet minimum region (purple curves). This is consistent with a larger computed k_NA (1.0 × 10⁻⁷) vs k_rot (3.2 × 10⁻¹⁰) value for 2N (Table 1). Compound 1N exhibits increased delocalization in both the singlet and triplet states at the rotational transition state and triplet minimum region, however, the effect of energy lowering is more pronounced for the singlet state (orange curves). And thus, the adiabatic and nonadiabatic rotational mechanisms become competitive. This is consistent with the computed k_NA (1.2 × 10⁻⁴) vs k_rot (4.6 × 10⁻⁴) values for 1N being of the same order of magnitude (Table 1). This comparison further suggests that more spin localized azoarenes likely undergo exclusive nonadiabatic rotational thermal isomerization, while more spin delocalized azoarenes likely undergo a combination of adiabatic and nonadiabatic thermal isomerization.

Compounds 3, 4, and 5 likely undergo Z–E isomerization via an exclusive nonadiabatic rotation pathway. Like 1N and 2N, the placement of NCH₃ groups around the bis-pyrazole rings in 3, 4, and 5 affects the degree of triplet spin delocalization, which correlates inversely to the M₁ and M₂ energies. Compound 3 is analogous to 1N, and 4 and 5 are analogous to 2N. Computed averaged spin densities at the diazo N atoms increase in the order: 3 (0.67) (most spin delocalized) < 4 (0.75) ≈ 5 (0.74). Accordingly, the computed M₁ and M₂ energies increase in the order: 3 (M₁ = 21.9 kcal/mol, M₂ = 20.6 kcal/mol) < 4 (M₁ = 25.4 kcal/mol, M₂ = 23.0 kcal/mol) < 5 (M₁ = 26.1 kcal/mol, M₂ = 24.2 kcal/mol) (Table 1) which agree qualitatively with the reported experimental thermal half-lives of 3 (0.3 days), 4 (72 days), and 5 (681 days).³² The structures of 3, 4 and 5 were computed based on the most stable conformers in DMSO, reported by He et al., determined by comparing the relative Gibbs free energies across a Boltzmann-weighted ensemble of possible conformers.³²

Substituent type and position can also influence the T₁ state energy, as illustrated by phenylazothiophene, 6. Resonance structures of the triplet state show that spin preferably reside at the ortho and para positions of the phenyl ring and at the C3 and C5 positions of the thiophene ring (Figure 2a). Consequently, introducing π-accepting (e.g., CN) or π-donating (e.g., OCH₃) groups at these positions is expected to stabilize the T₁ state. Conversely, placing such substituents at other positions, such as the meta position of the phenyl ring, would destabilize the T₁ state.

Compounds 6, 7p, 7m, 8p, and 8m likely undergo Z–E isomerization via a combination of adiabatic rotation and nonadiabatic rotation pathways. The computed M₁ and M₂ energies for 7p vs 7m and 8p vs 8m agree qualitatively with the trends of the experimental thermal half-lives of 7p (t_1/2 ~ 0.5 h) vs 7m (t_1/2 ~ 7 h) and for 8p (t_1/2 ~ 6.4 h) vs 8m (t_1/2 ~ 17.7 h).¹² In the −OCH₃ series, the computed average spin densities at the diazo N atoms increase in the order: 7p (0.63) (more spin delocalized) < 7m (0.66) (less spin delocalized) and the computed M₁ and M₂ energies increase in the order: 7p (M₁ = 22.1 kcal/mol, M₂ = 20.1 kcal/mol) < 7m (M₁ = 23.3 kcal/mol, M₂ = 21.6 kcal/mol) (see Table 1). In the −CN series, computed spin densities at the diazo N atoms increase in the order 8p (0.64) (more spin delocalized) < 8m (0.65) (less spin delocalized) and the computed M₁ and M₂ energies increase in the order: 8p (M₁ = 23.7 kcal/mol, M₂ = 20.3 kcal/mol) < 8m (M₁ = 24.5 kcal/mol, M₂ = 22.1 kcal/mol) (see Table 1).

Nonadiabatic processes (i.e., intersystem crossing) in photochemistry are relatively common and well-studied. When the reaction proceeds from a high-energy excited state, the minimum energy crossing point (MECP) is typically located at a lower energy and reaching it does not require thermal activation. The processes that promote the subsequent spin change—primarily spin–orbit coupling (SOC)—are temperature-independent (neglecting vibronic coupling effects). Thus, to a first approximation, such photochemical reactions are generally largely temperature-independent.

In the present case of azoarenes, the energetic situation is reversed: the reactions proceed from the singlet ground state and pass through an MECP that lies at a higher energy. Therefore, thermal activation is required, resulting in a temperature dependence of the nonadiabatic reaction, even though the spin transition itself remains mostly temperature-independent.

We have now investigated this temperature dependence for both the classical adiabatic and the triplet-assisted nonadiabatic processes in selected examples. In general, the adiabatic Z–E isomerizations are more sensitive to temperature changes than the nonadiabatic ones. For reactions such as those of compounds 7p, 7m, and 8p, in which both mechanisms are likely to occur simultaneously, the proportion of the classical adiabatic pathway increases with temperature. In parent azobenzene (Azb), the Z-isomer would need to be heated over 530 K for both processes to proceed at equal rates (Figure 3a). While in compound 7p, both mechanisms are already competitive at 330 K (Figure 3b). Thus, when discussing the reaction mechanism, temperature must be considered as a critical reaction parameter.

Figure 3.

Temperature dependence of ln(k_NA) (red) and ln(k_rot) (black) for compounds a) Azb and b) 7p.

Triplet spin density at the diazo linker nitrogen atoms emerges as a strong predictor for the trends of thermal Z–E isomerization half-lives in azoarenes. A good linear correlation was observed when experimental rate constants, ln(k_exp), were plotted against the computed average spin densities at the linker nitrogen atoms for triplet Azb, 2S, 3, 4, 5, 6, 7p, 7m, 8p, and 8m (Figure 4, R² = 0.870). Lower spin density at the linker nitrogen atoms indicates greater triplet spin delocalization, and correlates with less negative ln(k_exp) values, i.e., shorter thermal half-lives. Multilinear regression analysis gives a high correlation (R² = 0.910, Table S11) when average spin densities at the linker nitrogen atoms are modeled as a function of computed M₁ and SOCME values. This provides a physical basis for the observed relationship between ln(k_exp) and triplet spin density at the diazo nitrogens. In most cases, triplet spin resides predominantly on the diazo linker, preserving Hückel [4n+2] aromaticity in the arene rings. However, azoarenes with more spin-delocalized triplet spin exhibit reduced aromatic character,⁴⁰ as quantified by computed nucleus-independent chemical shifts (Figure S9).

Figure 4.

Correlation between ln(k_exp) vs computed averaged NBO spin densities at the linker N atoms for the T₁ minima geometries of Azb, 2S, 3, 4, 5, 6, 7p, 7m, 8p, and 8m.

CONCLUSIONS

Parent azobenzene is not the only compound that undergoes Z–E isomerization via a nonadiabatic mechanism, and some derivatives can undergo a combination of adiabatic and nonadiabatic rotations. We anticipate that the relationship between spin delocalization and thermal relaxation rates may also extend to other molecular solar thermal energy conversion systems (MOST) involving triplet state crossings, such as the [2 + 2]cycloreversion of quadricyclane to norbornadiene.⁴¹ In many pericyclic reactions that proceed at elevated temperatures, the initial formation of singlet diradicals is discussed (e.g., [2 + 2] cycloreversion of dioxetanes or the Cope rearrangement of 2,5-diphenyl-1,5-hexadiene). The findings presented in this study underscore that two-state or multispinstate reactivity is not exclusive to photochemistry, but may represent a key mechanistic principle in thermally driven transformations of organic molecules.

COMPUTATIONAL METHODS

Geometry optimizations, frequency calculations, and natural bond orbital (NBO)⁴² calculations were performed at PBE0-D3(BJ)⁴³/def2-TZVPP⁴⁴ employing Gaussian 16, Revision C.01.⁴⁵ Vibrational frequency analysis verified the nature of the minima and transition state structures. All geometries were optimized using a broken symmetry (BS) approach with the guess = mix keyword at the UPBE0-D3(BJ)/def2-TZVPP level. Minimum energy crossing point (MECP) optimizations were performed using ORCA 5.0⁴⁶ at the RIJCOSX⁴⁷-(BS)PBE0⁴³-D3(BJ)/def2-TZVPP⁴⁴ level using a tight SCF convergence criteria (VeryTightSCF) to locate the S₀ to T₁ crossing points (M₁) and the T₁ to S₀ crossing points (M₂). Since broken symmetry solutions are known to underestimate the rotational barrier for azobenzene,²⁵ single-point CASPT2/ANO-RCC-VTZP energies were computed for the E form minima and Z form minima, the M₁ and M₂ crossing points, as well as the rotation and inversion transition state structures to provide a more robust energy description.^24,26 CASPT2 calculations were performed using OpenMolcas Version 24.06⁴⁸ and details for the selected active space as well as energies can be found in the Computational Details section of the SI. Vibrational frequency and spin–orbit coupling calculations were computed at the optimized MECPs and used to derive the estimated nonadiabatic rate constants (k_NA), following the Wentzel–Kramers–Brillouin (WKB) theory (see Rate Constants and Thermochemistry Section in the SI). Temperature dependent k_NA and k_rot values were derived by the Eyring transition state theory (see SI for details) over a temperature range of 228 to 528 K, in 10 K increments, for Azb and 7p.

Supplementary Material

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacs.5c09852.

• Optimized Cartesian coordinates for all structures. Computational details for spin–orbit coupling matrix element (SOCME) calculations, vibrational analysis, computed rate constants and thermochemistry data, statistical analyses on the relationships between M₁, SOCME, ΔE^‡_inv and ln(k_exp), complete spin density data, computed nucleus-independent chemical shifts, and supplementary rotational potential energy scans. (PDF)